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%I #10 May 08 2017 00:20:43
%S 1,1,6,20,65,190,571,1616,4555,12439,33515,88517,230738,592321,
%T 1502384,3763946,9328899,22880511,55585077,133806273,319373068,
%U 756124040,1776497540,4143489680,9597505006,22083821765,50494638926,114758996621,259303832735,582655202940,1302234303910,2895530963661,6406348746390
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).
%C Euler transform of the square pyramidal numbers (A000330).
%H Vaclav Kotesovec, <a href="/A279215/b279215.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).
%F a(n) ~ exp(Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(24883200000*Zeta(5)^3) + Pi^8*Zeta(3)/(1728000*Zeta(5)^2) - Zeta(3)^2/(720*Zeta(5)) + Zeta'(-3)/3 + (Pi^12/(43200000*2^(3/5)*Zeta(5)^(11/5)) - Pi^4*Zeta(3) / (3600*2^(3/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(144000*2^(1/5)*Zeta(5)^(7/5)) + Zeta(3)/(12*2^(1/5)*Zeta(5)^(2/5))) * n^(2/5) + Pi^4/(180*2^(4/5)*Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5)/2^(7/5) * n^(4/5)) * Zeta(5)^(23/225) / (2^(29/150) * sqrt(5*Pi) * n^(271/450)). - _Vaclav Kotesovec_, Dec 08 2016
%t nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000330, A000335, A279216, A279217, A279218, A279219.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 08 2016